Existence of a solution to an infinite system of weighted fractional integral equations of a function with respect to another function via a measure of noncompactness

Definition 1.1

A map μ : ℬ ℋ → [ 0 , ∞ ) will be an MNC in ℋ if:

1. μ ( Ł ) = 0 deduces Ł is relatively compact for all Ł ∈ ℬ ℋ .

2. ker μ = { Ł ∈ ℬ ℋ : μ ( Ł ) = 0 } ≠ ∅ and ker μ ⊂ N ℋ .

3. Ł ⊆ Ł 1 ⇒ μ ( Ł ) ≤ μ ( Ł 1 ) .

4. μ ( Ł ¯ ) = μ ( Ł ) = μ ( Conv Ł ) .

5. μ ( δ Ł + ( 1 − δ ) Ł ) ≤ δ μ ( Ł ) + ( 1 − δ ) μ ( Ł ) for all δ ∈ [ 0 , 1 ] .

6. ℬ ℋ has the closed decreasing limit infinite intersection property, i.e., the intersection over any infinite closed decreasing subcollection of ℬ ℋ is non-empty, where lim j → ∞ μ ( Ł j ) = 0 .

Theorem 2.1

Let ℒ be an NBCC subset of a Banach space ℋ ¯ . Also, let K : ℒ → ℒ be a continuous mapping which is α -admissible with α ( ℒ ) ≥ 1 . Also,

for all nonempty £ ⊆ ℒ , where μ is an arbitrary MNC, A ∈ A , ℵ ∈ Φ and ℏ ∈ Ψ ℵ . Also, assume that γ : R + → R + is an increasing and continuous mapping. Then K possesses at least one fixed point in ℒ .

Proof

Consider the sequence { ℒ p } p = 1 ∞ with ℒ 1 = ℒ and ℒ p + 1 = Conv ( K ℒ p ) for all p ∈ N . Also, K ℒ 1 = K ℒ ⊆ ℒ = ℒ 1 , ℒ 2 = Conv ( K ℒ 1 ) ⊆ ℒ = ℒ 1 . Continuing in the similar manner gives ℒ 1 ⊇ ℒ 2 ⊇ ℒ 3 ⊇ … ⊇ ℒ p ⊇ ℒ p + 1 ⊇ … .

On the other hand, as K is α -admissible and α ( ℒ ) ≥ 1 , so, α ( ℒ 2 ) = α ( Conv K ℒ 1 ) ≥ 1 .

Using mathematical induction, we obtain α ( ℒ p ) ≥ 1 for all p ≥ 1 .

Now, for all p ∈ N ,

If there exists p 0 ∈ N satisfying μ ( ℒ p 0 ) = 0 , then ℒ p 0 is a compact set. In this case, the Schauder’s FPT implies K has a fixed point in ℒ .

Now, let μ ( ℒ p ) > 0 , for all p ∈ N . Since μ ( ℒ p + 1 ) ≤ μ ( ℒ p ) , for all p ∈ N , consequently, there exists l ≥ 0 such that lim p → ∞ μ ( ℒ p ) = l .

If l = 0 , then K has a fixed point.

If l > 0 when p → ∞ we obtain that

which is a contradiction. Consequently, l = 0 , i.e., lim p → ∞ μ ( ℒ p ) = 0 .

Since ℒ p ⊇ ℒ p + 1 , by Definition 1.1, we obtain ℒ ∞ = ⋂ p = 1 ∞ ℒ p is an NBCC subset of ℒ and ℒ ∞ is T invariant. Thus, according to theorem 1.2, K admits a fixed point in ℒ . □

Corollary 2.2

For α ( £ ) = 2 a , a ≥ 0 ; ℏ ( t ) = 2 t 2 ; ℵ ( t ) = t 2 and A ( x , y ) = x + y condition (2) reduces to

For α ( £ ) = K , K ≥ 1 ; ℏ ( t ) = 3 t 3 ; ℵ ( t ) = t 3 and A ( x , y ) = x + y condition (2) reduces to

Theorem 2.3

Let ℒ be an NBCC subset of a Banach space ℋ . Also, let K : ℒ → ℒ be a continuous and α -admissible mapping with α ( ℒ ) ≥ 1 . Also,

for all nonempty £ ⊆ ℒ , where μ is an arbitrary MNC, ℵ ∈ Φ and ℏ ∈ Ψ ℵ . Also, let γ : R + → R + be an increasing and continuous mapping. Then K possesses at least one fixed point in ℒ .

Proof

The result follows by taking A ( ℏ , ℘ ) = ℏ + ℘ in Theorem 2.1.□

Theorem 2.4

Let ℒ be an NBCC subset of a Banach space ℋ . Also, let K : ℒ → ℒ be a continuous mapping and

for all nonempty £ ⊆ ℒ , where μ is an arbitrary MNC, A ∈ A , ℵ ∈ Φ , and ℏ ∈ Ψ ℵ . Also, let γ : R + → R + be increasing and continuous. Then K admits at least one fixed point in ℒ .

Proof

The result follows by taking α ( £ ) = 1 in Theorem 2.1.□

Remark 1

In Theorem 2.4, let A ( a , b ) = a + b , γ ( t ) = 0 , ℏ ( t ) = t and ℵ ( t ) = k t for all t ≥ 0 where k ∈ [ 0 , 1 ) . So,

So, we obtain the DFPT. Hence, our results generalize the DFPT.

Definition 2.5

[15] An element ( p , q ) ∈ Q × Q is called a coupled fixed point (CFP) of a mapping K : Q × Q → Q if K ( p , q ) = p and K ( q , p ) = q .

Theorem 2.6

[10] Suppose that ω 1 be an MNC in ℋ 1 , ω 2 be an MNC in ℋ 2 , … and ω n be an MNC in ℋ n . Moreover, let the map A : R + n → R + be convex and A ( ς 1 , ς 2 , … , ς n ) = 0 if and only if ς l = 0 for l = 1 , 2 , … , n . Then ω ( Q ) = A ( ω 1 ( Q 1 ) , ω 2 ( Q 2 ) , … , ω n ( Q n ) ) defines an MNC in ℋ 1 × ℋ 2 × ⋯ × ℋ n , where Q l denotes the natural projection of Q into ℋ l for l = 1 , 2 , … , n .

Example 1

[10] Let ω be an MNC on ℋ . Define A ( p , q ) = p + q , p , q ∈ R + . Then A has all the properties mentioned in Theorem 2.6. Hence, ω cf ( Q ) = ω ( Q 1 ) + ω ( Q 2 ) is an MNC in the space ℋ × ℋ , where Q l , l = 1 , 2 denote the natural projections of Q .

Theorem 2.7

Let ℒ be an NBCC subset of a Banach space ℋ , K : ℒ × ℒ → ℒ be a continuous mapping such that

for any nonempty Q 1 , Q 2 ⊆ ℒ , where μ be an arbitrary MNC and A ∈ A . Also, let γ : R + → R + be an increasing and continuous mapping, γ ( ℏ + ℘ ) ≤ γ ( ℏ ) + γ ( ℘ ) and ℏ ( ℏ + ℘ ) ≤ ℏ ( ℏ ) + ℏ ( ℘ ) . Then K has at least one CFP in ℒ × ℒ .

Proof

We observe that μ cf ( Q ) = μ ( Q 1 ) + μ ( Q 2 ) is an MNC on ℋ × ℋ for any bounded subset Q ⊆ ℋ × ℋ , where Q 1 , Q 2 denote the natural projection of Q .

Consider a mapping K cf : ℒ × ℒ → ℒ × ℒ by K cf ( u , v ) = ( K ( u , v ) , K ( v , u ) ) .

It is trivial that K cf is continuous. Let Q ⊆ ℒ × ℒ . We obtain

By Theorem 2.1, we conclude that K cf admits at least one fixed point in ℒ × ℒ , i.e., K has at least one CFP.□

Corollary 2.8

For ℏ ( t ) = 2 t 2 ; ℵ ( t ) = t 2 and A ( x , y ) = x + y the condition in Theorem 2.7  reduces to

For h ( t ) = 3 t 3 ; ℵ ( t ) = t 3 and A ( x , y ) = x + y the condition in Theorem 2.7  reduces to

Theorem 3.1

Under hypotheses (1)–(5), equation (5) has at least one solution in C ( J , ℓ p α ) where p > 1 .

Proof

For arbitrary fixed ℏ ∈ J ,

Therefore,

which implies